Write a polynomial function with integral coefficients having the given roots.
1.) 0, -1/2, 6
2.) +or- 5i
To write a polynomial function with integral coefficients having given roots, we will use the fact that if a number "a" is a root of a polynomial function, then (x - a) is a factor of the polynomial.
1.) Given roots: 0, -1/2, 6
To find a polynomial with these roots, we can use the factors:
(x - 0) = x, (x + 1/2), and (x - 6).
To form a polynomial with integral coefficients, we can multiply these factors together:
P(x) = x * (x + 1/2) * (x - 6)
Multiplying these factors, we get:
P(x) = x(x^2 - 6x + 1)
Hence, a polynomial function with integral coefficients having roots 0, -1/2, and 6 is P(x) = x^3 - 6x^2 + x.
2.) Given roots: ±5i
For complex roots of the form "a ± bi," both "a + bi" and "a - bi" will be roots of the polynomial.
So, the given roots ±5i imply that both 5i and -5i are roots.
To form a polynomial with integral coefficients, we can use the factors:
(x - 5i) and (x + 5i).
Multiplying these factors, we get:
P(x) = (x - 5i)(x + 5i)
Using the difference of squares identity (a^2 - b^2 = (a + b)(a - b)), this simplifies to:
P(x) = (x^2 - (5i)^2)
Simplifying further, we have:
P(x) = x^2 + 25
Hence, a polynomial function with integral coefficients having roots ±5i is P(x) = x^2 + 25.